Nature’s Trademark – Phi

A review report on the occurrence of the golden ratio in nature

Photo © Rahul Gupta

Rahul Gupta (grahul@iitk.ac.in) Kshitij Saxena (ksaxena@iitk.ac.in)

© Biological Sciences and Bio Engineering IIT Kanpur

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Abstract

Nature has engineered many of its parts based on the golden ratio(phi). Phi is the ratio of consecutive Fibonacci Numbers as the series tends to infinity. Be it the seeds in a sunflower, or the ratio of the length of our body to that of our torso, or the structure of the DNA- surprisingly they all yield a single number "Phi" or 1·61803398874989484820... In the report we explore this Holy grail of Nature’s design and also try and propose reasons as to why is this number so basic in the architecture of nature.

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Index Introduction- An overview of Phi In the DNA- Occurrence of Phi in THE molecule of life In Cell division- Occurrence of Phi in a process common to all living entities In the Cell structure- Occurrence of Phi in the vessel of life In the Human Body- Occurrence of Phi in the greatest creation Why Phi?- Simple question, but difficult answers Geometrical structures exhibiting Phi- Looking around to see the ocean of Phi

Phi in Plants- Strongest evidence of Phi

Why Phi again?- Simple question, better answers Conclusion- Summing up the Holy grail of Nature’s design

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Nature’s Trademark - Phi Introduction Does Nature play dice? It appears not. Nature has its set of biases in designing its components. Quite interestingly Nature’s bias is heavily tilted towards one single number, phi i.e. 1·61803398874989484820... This number appears in almost everything ranging from sunflowers to sea shells to Drosophila to the human body- it is probably Nature’s trademark. It is called the golden ratio. So what is it that makes phi so special? We need to explore its properties to know more. Phi is the ratio of consecutive numbers in the Fibonacci series as it tends to infinity. In the Fibonacci series each number is the sum of two numbers preceding it. ( 0,1,1,2,3,5,8,13….) . There are several other methods to come to the number Phi

Phi has strange mathematical properties too. There are just two numbers that remain the same when they are squared namely 0 and 1. Other numbers get bigger and some get smaller when we square them. Remarkably phi is also defined as :

Two quantities are said to be in the golden ratio, if "the whole (i.e., the sum of the two parts) is to the larger part as the larger part is to the smaller part", i.e. (a+b)/a = a/b. Quite interestingly this ratio comes out to be phi.

Phi2 = Phi + 1

Many other interesting properties are mentioned in Table 1. These facts give us a certain degree of insight into the golden ratio and will be instrumental for us to decipher the information stored by nature in this special number. But before probing the number further we shall look into various examples where this number shows up and in the process we shall try an uncover the mystery surrounding phi.

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In the DNA The DNA molecule, the program for all life, is based on the Golden section. It measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral.34 and 21, of course, are numbers in the Fibonacci series and their ratio, 1.6190476 closely approximates Phi, 1.6180339. It is interesting to note as the technology improves and we get more accurate dimensions the ratio keeps getting closer to phi. It now appears that the ratio of the width to the vertical offset may also converge to the same ratio.

Figure 1 Figure 2 Figure 2 shows how a cross-section of the DNA perfectly fits into a decagon formed of golden elements. The ratio becomes apparent in the frequency of bases in the DNA. Selvam (2002) using statistical tools has shown that the frequency of A,T,C,G bases in the Drosophila genome. His paper reads: “The observed fractal frequency distributions of the Drosophila DNA base sequences

exhibit quasicrystalline structure with long-range spatial correlations or self-organized

criticality… The dominant peak periodicities are functions of the golden mean.”

This way non-reductionist studies are opening new avenues for science to explore.

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In Cell division After having seen the golden ratio in the DNA, the basis of life on earth it will be interesting to see if this ratio is translated from the DNA structure into the cell cycle. Let us start with a mother cell A0. In cycle 1 there is just one cell A0. In cycle2, A0 during the mitosis duplicates into two daughter cells 2 A1.In cycle 3, the two mother cells, 2 A1, duplicate into four daughter cells: 4 A2 In cycle n, the 2n-2 An-2 cells, duplicate into 2n-1 daughter cells: 2n-1 An-1 . The number sequence which represents the cell division is a geometrical series:

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, As we have seen, by geometrical model of cell division, each cycle leads to the new cells of the same generation without surviving of the old cell. The cell A0 becomes older by the law

t=n*T where n is the number of the cycle and T is the period of of the cycle. But, by the previous model the age of the cells is always T. That is in contradiction with the natural law and with the reproduction of the multi cellular organisms. When we have reproduction of the multi-cellular organisms the parents kept their generation and the children represent a new generation. This implies that Cell division results into two cells of different age! We recognize them as the mother cell and the daughter cell. So in cycle 2, A0 divides for the first time and produces her first daughter cell: A0 + A1 In cycle 3, the mother cell A0 reproduces into A0 + A1, as well as cell-daughter reproduces into A1 + A2 . Now, three generations are present: A0 + 2 A1 + A2. In cycle 4, the original mother cell produces another daughter cell. Two mother cells A1 reproduce into 2 A1 + 2 A2. The mother cell A2 also produces its own daughter cell. Now four generations are present:A0 + 3 A1 + 3 A2 + A3 ; The number of cells in each cycle of cell division is : 1,3,4,7,11,18,29,47,76 That is Lucas series of numbers. Again this series has the characteristic that each term is the sum of the two previous terms. Fibonacci series is also a particular kind of Lucas series wherein the first two numbers are 0,1. Interestingly ratio of consecutive terms of any Lucas series as it tends to infinity is phi, the golden ratio.

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In the Cell structure The ratio appears in certain very important structures of the cell. The hexagonal pattern of microtubules exhibits the Fibonacci feature and it is found that this pattern is made up of 5 right-handed and 8 left-handed helical arrangements.

It is curious, also, that the double otubules that frequently occur seem

normally to have a total of 21 columns of tubulin dimers forming the outside

undary of the composite tube - the next Fibonacci number. Koruga (1974) argues for a special efficiency in the case of Fibonacci-number-related structure of microtubules that may provide advantage in its function as a "information processor". There must indeed be some good reason

r this kind of organization in icrotubules, since although there is some riation in the numbers that apply to

eukaryotic cells generally,13 columns seems to be almost universal amongst mammalian microtubules.

micr

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View down a microtubule! The 5 + 8 = 13 spiral arrangement of the tubulins in this microtubule can be seen.

Imagine a microtubule slit along its length, and then opened out flat into a strip. We find that the tubulins are ordered in sloping lines which rejoin at the opposite edge 5 or 8 places displaced (depending upon whether the lines slope to the right or to the left).

Figure 3 In the Human Body If we look at the bigger picture now, the number can be seen in various human body parts like the lungs and the heart. Gibson et al (2003) conducted an anatomical study in 2093 people and demonstrated that the Fibonacci cascade appears in the distribution of coronary artery lesions in the human heart. In the study conducted the mean length of the artery was 15.3 cm. Figure 4 shows a comparison between the expected and the observed lesion.

Figure 4

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Similar studies by West et al.(1984) were performed on the branching network of bronchial tubes in the lungs of multiple mammalian species, including humans. The bronchial tubes repeatedly fork into two daughter structures of unequal length as shown in Figure 5. In humans, the group examined the forking of tubes for seven generations and found that the mean ratio between the branches was 0.62 ± 0.02 (mean ± SEM).

Figure 5

In the body various organs team up to form an entire individual only to show the ratio again. The M/m ratio in Figure 6 always comes out to be 1.618. The following have been observed to be in the golden ratio.

The distance between the finger tip and the elbow / distance between the wrist and the elbow. The distance between the shoulder line and the top of the head / head length, The distance between the navel and knee / distance between the knee and the end of the foot.

Figure 6 One such historical observation that the ratio of the total height in humans to the vertical height of the navel approximates the golden mean has been verified in a systematic study by Davis (1979).

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On a smaller scale this study was carried out by our team too. Our results are tabulated in Table 2.

Height (cm) Navel to Toe Distance (cm) Ratio 180 111 1.622 167 102 1.637 169 108 1.565 173 106 1.632 178 110 1.618 175 107 1.636 161.5 98 1.648 160 97.5 1.641 176 108 1.63 179 111 1.613 Average 1.624

The human face abounds with examples of the golden ratio. The head forms a golden rectangle (A rectangle whose sides are in the golden ratio) with the eyes at its midpoint. The mouth and nose are each placed at golden sections of the distance between the eyes and the bottom of the chin. The front two incisor teeth form a golden rectangle, with a phi ratio in the height to the width. The human ear follows the pattern of the logarithmic spiral which is intimately related to phi.

Table 2

Figure 7

The golden ratio has become an important tool for beauty analysis too. Computer generated masks based on phi have been created and are being in plastic surgeries etc. We downloaded one such Phi mask available on the net and used an image software to apply it to various faces and test its utility. Some of our experiments are shown in Figure. Apparently, the mask fits well on faces that are generally considered attractive.

Figure 8

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Why Phi? It is worth noting that the number keeps recursively occurring as we move from one level of complexity to another. This helps us draw an analogy between the Fibonacci numbers and Nature. Both are self forming and self similar. Unlike most common day science which is obsessed with the reductionist approach Phi gives an insight which is based on synergy. It shows us that random may actually not be random but hold many a secrets for us to explore. Now, evolution is believed to progress through random events. The Golden Number provides some remarkable insights. Wagner et al. (2003) have shown that the pattern of repeats in the genome sequence follows the Fibonacci-Cayley index. Their paper reads: “ This implies that essentially nothing but the rather obvious exponential growth of the

sequence length of the order of can be concluded from the Fibonacci-Cayley

model of repetitive sequence growth even if very simple mutations only are admitted

during the growth process.”

This shows that in spite of the random nature of insertion/deletion of repeats a definite pattern of growth is followed. Probably that is why random events have given rise to an ordered world today. Other studies carried out on evolution have yielded interesting results relating to Phi. Onody and Medeiros (1999) created mathematical age structured population model containing all the relevant features of evolutionary aging theories. Their model took into consideration beneficial and deleterious mutations, heredity, and arbitrary fecundity all managed by natural selection. They showed that fertility is associated with generalized forms of the Fibonacci sequence. But why Phi? From a morphogenetic viewpoint, the generation of complex, irregular structures based in part on principles of fractal self-similarity and Fibonacci proportionality may serve to minimize constructional error (Thompson 1963). The propensity for this ratio to appear in nature is also be because this ratio optimizes the efficiency of packing structures in a limited space in such a way that wasted space is minimized and the supply of energy of nutrients is optimized. This is elaborated in detail in our subsequent discussion of Phi occurring in the arrangement of seeds. Phi thus becomes a standing example of how coding information in different ways (in this case Mathematics) makes us discover the ways Nature operates. Mathematically Phi is considered to be the most irrational number and this again will help us understand that why nature chose ‘Phi’ as its favorite number. The irrationality of phi is shown under: As introduced,

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phi = (a + b) / a = a / b where a is the larger part and b is the smaller part

Equivalently, they are in the golden ratio if the ratio of the larger one to the smaller one equals the ratio of the smaller one to their difference, i.e. if

Dividing the numerator and denominator of the second fraction by b, we get

and replacing a/b by φ, we get

This becomes

or equivalently,

The solutions of this quadratic equation are

Since φ is positive, we have

(This is yet another way to ascertain the value of phi).

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Since √5 is irrational, it suffices to be a proof for the irrationality of phi. However hereunder we also show why phi is the most irrational number around. The quadratic obtained above can be written in the following form

which can be expanded to obtain a continued fraction for the golden ratio as follows:

This imparts the ‘most irrational’ nature to phi. While other irrational numbers like Л and e can be approximated with rational fractions like 22/7 etc., Phi cannot. The continued fraction nature gives an implicit understanding to why Phi keeps occurring at each level as we climb the ladders of complexity. Now, having observed how Phi is expressed mathematically, it will be interesting to see how it manifests in some of the more famous geometrical structures. Geometrical structures exhibiting Phi The golden ratio is crypted in a myriad of shapes we see around us everyday. The simplest to begin looking into is the Golden Rectangle. If we start with two small squares of size 1 next to each other and on top of both of these draw a square of size 2 (= 1 + 1) and touching both a unit square and the latest square of side 2 - so having sides 3 units long; and then another touching both the 2-square and the 3-square (which has sides of 5 units)…. And so on, we will get a figure where the size of squares follows the Fibonacci sequence. The figure follows.

Figure 9

We can continue adding squares around the picture; each new square having a side which is as long as the sum of the latest two square's sides. This set of rectangles whose sides

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are two successive Fibonacci numbers in length and which are composed of squares with sides which are Fibonacci numbers, are called the Fibonacci Rectangles.

Figure 10

Here is a spiral drawn in the squares, a quarter of a circle in each square. The spiral is not a true mathematical spiral (since it is made up of fragments which are parts of circles and does not go on getting smaller and smaller) but it is a good approximation to a kind of spiral that does appear often in nature. Such spirals are seen in the shape of shells of snails and sea shells. The spiral-in-the-squares makes a line from the centre of the spiral increase by a factor of the golden number in each square. So, points on the spiral are 1.618 times as far from the centre after a quarter-turn. In a whole turn the points on a radius out from the center are 1.618 * 4 = 6.854 times further out than when the curve last crossed the same radial line. Following is an image of the Nautilus sea shell exhibiting close correlation with Logarithmic spirals. In the pictures, we have an image of the Nautilus shell superimposed on a Logarithmic spiral – what a match!

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These images show the spiral curve of the Nautilus sea shell and the internal chambers that the animal using it adds on as it grows. The chambers provide buoyancy in the water. If a line is drawn from the center out in any direction and intersects two places on the shell such that the shell spiral has gone round just once between the intersection points, then, the outer crossing point will be about 1.6 times as far from the centre as the next inner intersection point. This shows that the shell has grown by a factor of the golden ratio in one turn. Modelling of sea shells: The Nautilus sea shells adhere so much to the Logarithmic spirals that the actual 3D models of sea shells may be made using mathematical equations. Similarly, other sea shells may be modelled based on similar mathematical principles. A few images showing such models follow.

Mathematical Models of the Nautilus shell

Figure 12

Mathematical models of other shells

Examples of curves based on the logarithmic spiral can be seen in the tusks of elephants and the now-extinct mammoth, lions' claws and parrots' beaks. The eperia spider always

Figure 13

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weaves its webs in a logarithmic spiral. The spiral revolutions come closer together as they approach the pole. Such a pattern can also be seen in human fingerprints. Among the micro-organisms known as plankton, the bodies of globigerinae, planorbis, vortex, terebra, turitellae and trochida are all constructed on spirals. Even sunflower seeds arrange there seeds in the shape of the logarithmic spiral. This is discussed in detail in a following section.

Figure 14

Figure 15

Using the Phi Matrix Software, we divided the rectangle surrounding the moth in the golden ratio. It can be seen that many essential parts of the

moth fall at the points of intersection of the golden lines.

However, let us currently focus on more manifestations of Phi in simple geometrical structures. One of such structures is the dodecahedron. It consists of 12 pentagonal faces, and the icosahedron of 20 triangles. These shapes can all mathematically turn into one another, and that this transformation takes place with ratios linked to the golden ratio.

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Figure 16 For this we take three golden rectangles and assemble them at 90 degree angles to get a 3D shape with 12 corners as in Figure 16. The 12 corners become the 12 centers of each of the 12 pentagons that form the faces of a dodecahedron. The 12 corners can also become the 12 points of each of the 20 triangles that form the faces of a icosahedron.

Dodecahedron Icosahedron

The protein sheath of the Adeno virus consists of 252 protein subunits, all regularly set out. The 12 subunits in the corners of the icosahedron are in the shape of pentagonal prisms. Rod-like structures protrude from these corners.

Figure 17

The first people to discover that viruses came in shapes containing the golden ratio were Aaron Klug and Donald Caspar from Birkbeck College in London in the 1950s. The first virus they established this in was the polio virus. The Rhino 14 virus has the same shape as the polio virus.

Having seen the how Phi appears in basic geometrical structures, we are now in a better position to directly visualize the manifestations of Phi in more complex natural structures. So, let us go back to our sunflower problem and take it by the horns ☺

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Phi in Plants

In botany, phyllotaxis or phyllotaxy is the arrangement of the leaves on the shoot of a plant. It may also refer to the arrangement of flowers or buds on a stem or to the arrangement of seeds on a flower. Basically, phyllotaxis refers to the morphological arrangement of plant parts on supporting structures. There are four main types of phyllotaxis found in nature. These are: 1) Distichous Phyllotaxis 2) Whorled Phyllotaxis 3) Spiral Phyllotaxis and 4) Multijugate Phyllotaxis. Of these, we are primarily interested in spiral phyllotaxis and multijugate phyllotaxis only.

Spiral Phyllotaxis (Figure 18)

In spiral phyllotaxis, botanical elements grow one by one, each at a constant divergence angle ‘d’ from the previous one. This is the most common pattern, and most often the divergence angle d is close to the Golden Angle, which is about 137.5 degrees. The latter case gives rise to Fibonacci Phyllotaxis. It is interesting to see how this angle is derived. Vogel(1979) theorized that primordia, no matter what they developed into, could fill space most efficiently when the divergence angle was an irrational part of 360. Most intuitively if we divide 360 by 1.618 we get 222.5. This angle is the reflex angle and subtracting it from 360 would give 137.5

Figure 19

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Multijugate phyllotaxis (Figure 20)

In multijugate phyllotaxis, two or more botanical elements (two in the example above) grow at the same node. Elements in a whorl (group of elements at a node) are spread evenly around the stem and each whorl is at a constant divergence angle d from the previous one. Often, multijugate patterns look very similar to spiral patterns and the only way to detect them is to count the number of spirals visible in the pattern. Counting Spirals: To further classify spiral and multijugate patterns, one counts the number of visible spirals, called parastichies, which join each element to its nearest neighbors. These spirals normally come in two families, yielding a pair of numbers, called parastichy numbers. If the parastichy numbers have no common divisor other than 1, the pattern is a spiral phyllotaxis. If the parastichy numbers do have a common divisor k, then the pattern is multijugate (more precisely k-jugate) and there are k elements at each node. If the parastichy numbers are (k, k), then, we get what is called the “whorled” phyllotaxy.

Figure 21 The number of visible spirals gives the parastichy numbers. The number of spirals clockwise / anti-clockwise together gives one set of parastichy numbers. The Aonium in Fig 20 has parastichy numbers (2, 3). Since 1 is the only common divisor of 2 and 3, this is a spiral pattern. Since spiral phyllotaxis can be viewed as 1-jugate, the notation 1(2, 3) is also used for this pattern.

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Figure 22 The gymnocalycium in Fig 21 has parastichy numbers (10, 16), which have the common divisor k=2. Hence this is a multijugate pattern (more precisely 2-jugate). The notation 2(5, 8) is also used to classify this pattern. The configuration is a Fibonacci phyllotaxis when the parastichy numbers are successive elements in the Fibonacci sequence. The case with identical parastichy numbers or k (1, 1) trivially belongs to Fibonacci phyllotaxis as (1, 1) are successive elements of the Fibonacci sequence. This is the case of whorled phyllotaxis. Hence, all cases of whorled phyllotaxis are also cases of Fibonacci phyllotaxis. Fibonacci phyllotaxis has the unique property that the diversion angle approaches 137.5º - the golden ratio angle.

Mammilaria huitzilo

Clockwise: 13 spirals.

Anti-Clockwise: 21 spirals. Parastichy numbers: (13, 21).

Spiral Phyllotaxis, Fibonacci Phyllotaxis (Figure 23)

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Marguerite

Clockwise: 21 spirals. Anti-Clockwise: 34 spirals.

Parastichy numbers: (21, 34). Spiral Phyllotaxis, Fibonacci Phyllotaxis

(Figure 24)

Knautia arvensis

Clockwise: 10 spirals. Anti-Clockwise: 6 spirals.

Parastichy numbers: 2 (3, 5). Multijugate Phyllotaxis, Fibonacci Phyllotaxis

(Figure 25)

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Relative Frequency of Patterns: Spiral phyllotaxis is the most common kind of phyllotactic pattern. According to a survey by Hutchinson, in a sample of 650 dicotyledonous families, 50.6% showed exclusively spiral or multijugate phyllotaxis, 14.7% showed exclusively whorled phyllotaxis while 34.7% showed a mixture of these phyllotaxis. According to a compilation by Jean (1994) of many surveys of 650 species and 12500 specimens, among plants showing spiral or multijugate phyllotaxis, about 92% showed Fibonacci phyllotaxis (with parastichy numbers successive elements in the Fibonacci sequence). Why Phi again? This is the first question that comes to the mind when we realize that 92% of all plants exhibiting spiral or multijugate phyllotaxies exhibit Fibonacci phyllotaxis. The reason can be seen in a simple model of seed placement on a seed head. Let us consider a fraction ‘f’ of turns between two successive seed placements. Seeds then grow farther from the meristematic centre at a constant radial speed. If we use 0.5 turns per seed, Since 0·5=1/2 we get just 2 "arms" and the seeds use the space on the seedhead very inefficiently: the seedhead is long and floppy.

A circular seedhead is more compact and would have better mechanical strength and so be better able to withstand wind and heavy rain. Also, a circular seedhead seems more efficient in terms of supporting tissue utilization. 0.48 turns per seed,

The seeds seem to be sprayed from two revolving "arms". This is because 0·48 is very close to 0·5 and a half-turn between seeds would mean that they would just appear on alternate sides, in a straight line. Since 0·48 is a bit less than 0·5, the "arms" seem to rotate backwards a bit each time.

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0.6 turns per seed

0·6=3 / 5 so every 3 turns produce exactly 5 seeds and the sixth seed is at the same angle as the first, the seventh in the same (angular) position as the second and so on. 0.61 turns per seed,

All three are for 0.61 turn per seeds but with increasing number of total seeds This is significantly better. However, there are still large gaps between the seeds nearest the centre, so the space is not best used. In fact, any number which can be written as an exact ratio (a rational number) would not be good as a turn-per-seed angle. If we use p/q as our angle-turn-between-successive-turns, then we will end up with q straight arms, the seeds being placed every p-th arm. Hence, we should resort to the irrationals. ‘e (2·71828...)’ turns per seed

We notice that the e picture has 7 arms since its turns-per-seed is (two whole turns plus) 0·71828... Of a turn, which is a bit more than 5 / 7 (=0·71428...).

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‘pi’ turns per seed

A similar thing happens with pi (3·14159..) since the fraction of a turn left over after 3 whole turns is 0·14159 and is close to 1 / 7 = 0·142857.. . It is a little less, so the "arms" bend in the opposite direction to that of e's (which were a bit more than 5/7). Thus, even the irrationals are a bad choice for the seedhead because they can be approximated by a rational number. Hence, we need a number that cannot be approximated by a rational number. Hence, we resort to the theory of continued fractions. We need an irrational number that never settles down to a rational approximation for very long. The simplest such number is that which is expressed as P = 1+1/ (1+1/ (1+1/ (...) I.e. P = 1 + 1 / P, …….. (1) However, this is one of the very definitions of Phi as has been discussed before. On solving (1), we get its roots as Phi = (√5 + 1) / 2 and phi = (√5 – 1) / 2. This reason can be extended onto the arrangement of leaves on a shoot. Successive leaves must arrange themselves in such a way so as to intercept most of the sunlight without casting shadows on each other. Hence, diversion angle 137.5º, optimised packing - Fibonacci phyllotaxis! Thus, we realise that the primary reason that evolution has chosen Phi in vast and diverse forms is to optimise packing. This model of optimised packing can be used in a lot of situations to gain a better understanding into the design of nature, particularly, in understanding why nature chose Phi to represent some of the most interesting features of design. Now that we have understood why Phi occurs in Phyllotaxis, let us get to contemporary research. We began by observing the golden ratio in cell division. At present, the challenge is to correlate asymmetric patterns of cell division with the generation of Fibonacci patterns, and to design tests to distinguish between these models. Perhaps the most useful approach may be to study mutants with altered developmental patterns.

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Takahashi et al (2002) have shown that a mutation in the topoisomerase coding gene which plays a major role in DNA replication leads to the disruption of the Phi patterns in plants. Their paper reads: “….. Here, we show that disruption of an Arabidopsis topoisomerase (topo) I gene named TOP1α affects phyllotaxis and plant architecture. The divergence angles and internode lengths between two successive flowers were more random in the top1α mutant than in the wild type….. . These morphological abnormalities indicate that TOP1α may play a critical role in the maintenance of a regular pattern of organ initiation….” This shows that the phyllotactic occurrence of Phi may have very deep roots, deep inside the genetic codes of organisms. Conclusion All these evidences and several more that are yet to be discovered make the golden ratio an indispensable number. We began showing evidences for occurrence of the golden ratio in the DNA and cell structure. Towards the end we were able to show how certain changes in the DNA sequence actually disrupts the Phi pattern. Thus when information is transferred form the molecular level to the macro level, there is one thing that is conserved- Phi. The journey ahead will be interesting as we try and fill the gaps in the translation process. Phi may in future give rise to a new branch of non reductionist science where correlation between entities is far more important than understanding their discrete functions. Already, we have been able to show the strong correlation the genome sequence has with phi. A correlation was also be made between Fibonacci Numbers and the ways of operation of Nature. In Fibonacci numbers each number owes its identity to what has occurred before it. Thus the series is self forming. Similarly in nature operates on this very same principal of self organization. Hence Phi is a number which becomes conspicuous in both. Over the billions of years since nature has chosen phi as its trademark, efforts are on to device nanostructures wherein the particles are organized in this ratio so as to minimize the stresses. Several more inventions knock the horizon and are waiting to be unleashed. However a lot of reading between the lines of nature’s design is yet to be done. The journey to explore more about the holy grail of nature’s design will certainly be very exciting.

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References 1. Harel (2003): Beauty is in the genes of the beholder (200-204 DNA50)

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Popular sites for reference:

• http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html • http://www.beautyanalysis.com/index2_mba.htm • mathworld.wolfram.com/GoldenRatio.html • www.geom.uiuc.edu/~demo5337/s97b/art.htm • http://en.wikipedia.org/wiki/Golden_ratio • www.phimatrix.com/

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